Session 3565 
                
                           Calculus Reform, Differential Equations and Engineering 
                                                      Michael Ruane 
                                  Electrical & Computer Engineering, Boston University 
                
                
                
                
               Abstract 
                
               Through much of the 1990’s, the National Science Foundation supported the development of 
               new pedagogical methods, textbooks, and teaching materials, including software, for calculus.  
               This was done in response to concerns from mathematicians, and from scientists and engineers 
               in client disciplines who believed students were not being well prepared for further study with 
               calculus applications.  The ’calculus reform’ movement has significantly changed the teaching 
               and learning of calculus where it is implemented and has been a constant topic of discussion in 
               the mathematics community.  Calculus reform is often unknown among engineering faculty. 
                
               Three Boston University mathematics faculty developed a ’reform’ differential equations course, 
               textbook, and computer labs, incorporating an unusual degree of engineering applications, 
               modeling and jargon.  Their systems approach was later disseminated in workshops to the math 
               community.  In those workshops, a common theme from the math professors was ’we don’t talk 
               with the engineers--they don’t even know that we’re teaching differently!’ 
                
               Calculus reform, including the NSF differential equations project at Boston University, 
               emphasizes using graphical solutions, numerical solutions, and symbolic solutions, as well as 
               writing about mathematics, discovery learning, and team-based learning.  This talk will describe 
               these elements and discuss the possibilities for closer collaboration between mathematics and 
               engineering around reform of the curriculum. 
                
               Introduction 
                
               The early 1980’s saw growing discussions in the mathematics community about the role and 
               effectiveness of calculus instruction, particularly in the freshman year.  Concerns were diverse.  
               Some argued that discrete mathematics should become the core undergraduate mathematics 
               course.  Others felt the calculus sequence had lost sight of essentials under the burden of 
               covering an increasing list of topics demanded by client disciplines and publishers.  Changing 
               instructional technology and new appreciation for student learning models seemed to require 
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               new curricular approaches.    
                
               In January, 1986, a Tulane University Conference produced a report “Toward a Lean and Lively 
               Calculus”3 which attempted to start a complete redesign of single variable calculus pedagogy 
               and content.  Almost immediately a strong case was made for computer-based algebra and 
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               plotting tools to help students overcome widespread weaknesses in numerical and symbolic                 age 6.256.1
               Proceedings of the 2001 American Society for Engineering Education Annual Conference & Exposition 
               Copyright © 2001, American Society for Engineering Education 
                
               manipulation in calculus.  The Mathematical Association of America (MAA) quickly created a 
               committee on calculus reform to continue the Tulane Conference initiatives, and the National 
               Science Foundation proposed a major calculus reform initiative.  About the same time, the 
               National Research Council started a project ‘Mathematical Sciences in the Year 2000’ to define 
               a calculus for the new century.   By 1988 there were 43 NSF projects ($7M) underway in 
               calculus reform, from the level of community colleges to major research universities4.   Most 
               projects had some form of computer algebra system to free students from hand manipulations.  
               In return students were asked to address more complicated, realistic problems, to use graphing 
               tools to develop a geometric understanding of calculus, to reflect upon and write about their 
               mathematical work, to explore mathematical problems until they could create meaning and 
               understanding for themselves and to participate in some forms of team-oriented learning. 
                
               Early reform efforts frequently just added computer exercises to existing topics (similar to the 
               current 'add a CD in the back' approach to engineering textbook reform).   It became apparent 
               that computer exercises alone were merely a patch on the older system, and that a more 
               extensive restructuring of the curriculum was possible with the new computational tools.  In 
               particular, students could learn the concepts of calculus and immediately apply them to 
               complicated real problems with appropriate computational tools.  Later they would develop 
               symbolic and manipulative skills.  
                
               Reform efforts have not been universally embraced and strong critics have emerged.5,6  Charges 
               are made that students are 'cheated' by computer work at the cost of terse mathematical 
                                                                                  7,8
               derivations.  The continuing national debate on educational reform    standards-based education, 
               and Science, Math, Engineering, Technology (SMET) education has invoked calculus reform 
               both as a success story and as a misguided effort.  Unfortunately, engineers seem to be absent 
               from these discussions, although our students are the largest client group for calculus.   
                
               On-going Calculus Reform Activities 
                
               Calculus reform has not faded away, but it also has not been universally adopted.  The most 
               popular calculus reform textbook 9 has been adopted by over 500 institutions, and has even been 
               used widely for high school AP classes.  Developed by a consortium of faculty from 11 colleges 
               and secondary schools, this book has now spawned competitors who offer both traditional and 
               reform elements.  Multivariate calculus and differential equations have 'reform' texts. 
                
               Evaluations of the reform movement has been on-going with the NSF development grants, but 
               with mixed empirical results 10,11.   NSF is currently funding a Clemson University study (NSF 
               9912017) on long-term student performance under calculus reform.  Earlier studies focused on 
               calculus skills, and did not consider how learning experiences and strategies from reform 
               calculus might improve performance in other areas, e.g. computationally intensive engineering 
               curricula.  The Clemson study will look at performance in classes outside calculus. 
                
               The MAA provides a forum for continuing development and discussion of calculus reform.   
               MAA's Calculus Reform and the First Two Years (CRAFTY) Committee was involved with the                      P
               original Tulane Conference, and has continued to offer panels and symposia, especially at the               age 6.256.2
               Proceedings of the 2001 American Society for Engineering Education Annual Conference & Exposition 
               Copyright © 2001, American Society for Engineering Education 
                
               annual joint meetings of the American Mathematical Society (AMS), Mathematical Association 
               of America (MAA) Association for Women in Mathematics (AWM) and the National 
               Association of Mathematicians (NAM).  The Committee on the Undergraduate Program in 
               Mathematics (CUPM) of the MAA launched a new curriculum initiative in 1999, and is working 
               with CRAFTY to conduct a series of workshops to understand the mathematics needs of 
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               students in math intensive disciplines  . Called the Curriculum Foundation Workshops, these 
               eleven meetings have generated reports that can help initiate discussions between mathematics 
               departments and their colleagues in engineering, physics, computer science, business, health 
               sciences, statistics, biology, chemistry and math education.  Specific documents are available 
               addressing the mathematics needed to support study and practice in Chemical Engineering, Civil 
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               Engineering, Electrical Engineering and Mechanical Engineering  . 
                
               The Boston University Differential Equations Project 
                
               Three mathematics professors at Boston University developed a new text 14, accompanying 
               software, and a web site in order to ’rethink completely the traditional, sophomore-level 
               differential equations course’.  The development effort was supported by NSF, which also 
               supported a series of summer dissemination workshops and workshops at the Mathematics Joint 
               Meetings.  They have eliminated most specialized techniques and introduced more topics on 
               formulating and interpreting differential equations.  They use qualitative methods and extensive 
               computer tools for visualizing solutions, looking for eigenvalues and eigenvectors, and 
               examining the phase plane solution space.  Numeric methods are applied throughout. 
                
               The project incorporates modeling and draws examples from many disciplines, including 
               engineering.  Often these examples are revisited with new techniques and complexity.  A 
               dynamical systems perspective in central to the book, and solution behaviors are examined as 
               model parameters are varied.  Linear algebra is introduced as needed.  Students are regularly 
               expected to complete ’labs’ with extensive numerical experimentation, and to write about their 
               findings.  Team-based labs are common. 
                
               The dissemination workshops were directed towards mathematics professors, and highlighted 
               the systems approach and computational tools supporting the students’ investigations.  Ample 
               time was provided to apply the computer tools during the workshops.  Additionally, one 
               afternoon of each two-day workshop was devoted to engineering applications of differential 
               equations, using their modeling and computational tools.  Electronics laboratory space was 
               coordinated at each workshop to allow the mathematicians to construct RC first and second 
               order systems, measure time constants and physically tweak parameters in their equations.  The 
               textbook authors served as the teaching assistants in the electronics lab!  All survived the labs, 
               and appreciated more the jargon and context of differential equations for their engineering 
               students. 
                
               Opportunities for Collaboration 
                
               Collaboration in the Boston University Differential Equations Project began while the book was             P
               being drafted.  One of the authors (Blanchard) audited the engineering circuit theory course               age 6.256.3
               Proceedings of the 2001 American Society for Engineering Education Annual Conference & Exposition 
               Copyright © 2001, American Society for Engineering Education 
                 
                (usually taken by students in the same semester as differential equations) and observed how 
                engineering texts present mathematics, apply jargon, and emphasize modeling and interpretation 
                over solution techniques.  This experience colored many of the examples and even the topic 
                development used in the text.   
                 
                The author was the ’token engineer’ for the NSF dissemination workshops, and was able to hear 
                how different schools collaborated between mathematics and engineering or other client 
                disciplines.  Most reported little if any regular communication between engineering and math 
                faculty, until a crisis arose.  This might be a budget cut, a technology proposal, or simply an 
                outraged engineering professor reporting that the students were not being taught ’correctly’.  
                Usually the engineering professor did not exactly know what was being taught or how, just that 
                the end product was unacceptable.  Everyone agreed that this was not productive or collegial.   
                The mathematics faculty were often reluctant to ask for help, especially where there was fear that 
                client disciplines would institute ’just-in-time’ math and possibly threaten math teaching jobs. 
                 
                Some continuing collaboration has resulted.  As part of an NSF Combined Research and 
                Curriculum Development grant, engineering faculty at Boston University continue to visit the 
                differential equations course each semester to give an applications lecture, usually on the 
                relation of predator prey systems to laser dynamics.  The lecture engages the sophomores with 
                lasers and applications, and then uses the computer tools to vary laser cavity characteristics until 
                the equations give desired transient behavior.  The lecture is well received by the students, even 
                those outside engineering.  Unfortunately, on most days and topics, there remains little 
                connection between the faculty in math and engineering.  It has proven difficult to get faculty in 
                a research I institution to commit the time and effort for continuing collaboration, even when 
                they agree it would improve student learning. 
                 
                Calculus Reform and ABET Reform 
                 
                ABET Criteria 2000 curricular reform started later than calculus reform, and has a much broader 
                mandate.   But the two efforts share many goals.  Both have been shaped by the changing 
                student population, new instructional technologies, renewed emphasis on modeling, qualitative 
                understanding, applications, writing and team learning.  The NSF participated in many of the 
                ABET discussions, especially in defining the national needs for an educated technical 
                workforce.   
                 
                Calculus reform can be a cautionary tale for ABET efforts.  Systematic change has been sought 
                for over 15 years in calculus reform, and there is still not wide agreement as to the effectiveness 
                of the changes.  Some are opposing the reform movement, and trying to restore older methods.  
                The relentless pace of change in students preparation, technology, and client needs has initiated 
                new calls for reform of the reforms (e.g. the CUPM foundation workshops).  These difficulties 
                arose despite a focus on just the calculus.  Engineering, with its broader content, will be more 
                vulnerable to parochial battles within its sub disciplines.   
                 
                Calculus reform has sought to create curricular overhaul through its investment in entirely new              P
                textbooks, pedagogy, and instructional technology.  While there are some ’old calculus’ texts                age 6.256.4
                Proceedings of the 2001 American Society for Engineering Education Annual Conference & Exposition 
                Copyright © 2001, American Society for Engineering Education